基于不同先验获取的PET图像优质重建新方法研究
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摘要
作为一种有效的功能医学图像成像方法，正电子发射成像(Positron Emission Tomography，PET)能够在分子水平上利用影像技术反映人体心脑代谢和功能，已经在肿瘤学，心血管疾病学和神经系统疾病学研究中，以及新医药学开发研究等邻域中显示出它卓越的性能。可是在实际中由于受到低计数率和一些物理噪声的影响，PET图像的重建问题在理论上是一个病态的问题。传统的滤波反投影(Filtered Back-Projection，FBP)重建方法虽然具有成像速度快的优点，其重建图像却含有大量噪声，图像质量较差。
最大似然—期望最大法(Maximum-Likelihood Expectation-Maximization，ML-EM)能够针对系统模型的物理效应和探测数据和噪声的统计泊松特性建立数学模型，其重建的图像质量要优于传统的FBP方法。然而，单纯的传统ML-EM方法依然有两个缺点：1，在迭代过程中，会产生质量退化的图像而导致的棋盘效应，从而导致非收敛的迭代过程；2，收敛速度较慢，需要大量迭代次数才能重建出可以接受的图像。
近二十年来，国际上出现了很多解决以上两个问题的方法。一方面，基于马尔可夫随机场(Markov Random Fields，MRF)的贝叶斯(Bayesian)重建方法或者最大化后验估计(Maximum A Posteriori，MAP)的方法通过引入正则化项来引进目标同位素密度数据在空间上的概率分布的先验信息，能够明显改善重建图像质量以及迭代过程的收敛性，该方法已被证明了其在理论上的正确性和实际上的有效性。另一方面，很多科学家也提出一些有效的方法和算法来解决ML-EM方法收敛速度慢的问题，1994年，H．Malcolm Houdson和Richard S．Larkin提出使用有序子集(Ordered Subsets，OS)的方法将完整探测数据sinogram分割成有序部分数据的方法来减小每次迭代的运算量，从而达到了加快收敛速度的目的；密歇根大学的J．A．Fessler小组提出了能够提高收敛速度的SAGE(Space-Alternating Generalized EM)收敛重建算法和PSCD(Paraboloidal Surrogate Coordinate Ascent)收敛重建算法；另外，Erkan (?) Mumcuoolu，David S．Lalush及Fessler也分别提出将共轭梯度法(Conjugate Gradient，CG)应用于PET重建算法中以加快收敛速度。
对于重建速度的问题，虽然能够反映探测数据统计特性的迭代重建能够重建出较传统的FBP重建更好的图像，然而其应用一直受到重建速度慢的缺点的限制；在重建图像质量方面，作者发现，sinogram探测值中大量的噪声一直对重建有着很大的负面影响，而且此种负面影响持续贯穿于整个在迭代过程中。虽然引入了图像先验信息的Bayesian能够在很大程度上改善迭代重建，但是依赖于传统的局部邻域信息的Bayesian方法只能为重建提供有限的局部先验信息，一方面，传统的平滑二次QM(Quadratic Membrane)先验易在重建的结果图像中产生过平滑效果，另一方面，具有边缘保持作用的非二次先验则会给重建图像带来阶梯状的边缘伪影。
本文对于PET重建算法的研究工作同样也是基于如何进一步加快重建算法的收敛速度和进一步提高重建图像的质量。作者做了以下四项PET重建算法方面的工作：
1，提出将有序子集的思想和共轭梯度法相结合的新的快速子集共轭梯度重建算法(OSCG Ordered Subsets Conjugate Gradient)来加快PET重建的收敛速度。
2，提出基于修改sinogram探测值的新的耦合反馈(Coupled Feedback，CF)迭代模型来降低在重建迭代中探测数据sinogram中的噪声对PET重建的负面影响。
3，提出一种新的有效的综合了QM先验和QP(Quadratic Plate)先验的MRF二次Membrane-Plate混合先验模型，实现了在Bayesian重建中根据目标图像中各点的性质来自适应的选择QM先验和QP先验。
4，提出一个新的且具有二次先验能量形式的MRF非局部(nonlocal)先验模型，实现了利用图像中的全局的信息来为PET重建提供更加有效的正则化作用。
试验中将以上四种方法应用于相应的PET断层发射成像(Emission Tomography)和断层透射成像(Transmission Tomography)，相关试验分析表明：本文所提出的四种基于MRF和优化理论的四种新算法均能够分别在不同程度上改善PET重建。
As an effective functional imaging method for medical images, positron emission tomography (PET) is able to represent heart and brain metabolism and functions on molecule level by imaging techniques, and has shown great performance in oncology, cardiopathy, neurology and new medicine studies. But positron emission tomography is an ill-posed inverse problem because the observed projection data are contaminated by noise due to low count rate and physical effects. Though needing less compution cost, traditional filter back projection (FBP) method often reconstruct noisy images of low quality.
Better expressing system models of physical effects and modeling the statistical poisson character of the data, the famous maximum-likelihood expectation-maximization (ML-EM) approach outperforms the FBP method with regard to image quality. However, pure traditional ML-EM approach is still notorious for two drawbacks: 1, the reconstructed images always start deteriorating to produce "checkerboard effect" as the iteration proceeds; 2, ML-EM approach suffers slow convergence, and a large number of iterations are needed to be performed before obtaining acceptable reconstructions.
Many methods have been proposed to overcome above two frawbacks in the past twenty years. On one hand, Bayesian methods or equivalently MAP (Maximum A Posteriori) methods, which incorporate MRF prior information of objective isotope density data into the ML-EM algorithm through regularization or prior terms, have been proved theoretically correct and practically effective compared to other methods. Compared to traditional ML-EM algorithm, Bayesian reconstruction shows a better performance in both improving convergence behavior and producing more appealing images. On the other hand, as to the drawback of slow convergence for ML-EM algorithm, some effective solutions and algorithms have also been proposed. In 1994, H. Malcolm Houdson and Richard S. Larkin proposed segementing the original whole sonogram data into several ordered data by OS (Ordered Subsets) method to lower the compution cost of each iteration step; J. A. Fessler and his group proposed fast convergent SAGE (Space-Alternating Generalized EM) algorithm and PSCD (Paraboloidal Surrogate Coordinate Ascent) algorithm for PET. And Erkan (?) Mumcuoglu, David S. Lalush and Fessler also applied CG (Conjugate Gradient) in PET reconsruction algorithm to obtain fast convergence rate.
As to the problem of reconstruction rate, the iterative methods which incorporate the statistical characters of scaned data, although able to reconstruct better images than conventional FBP approaches, their applications are also hindered by the slow reconstruction rate. As to the problem of reconstructed image quality, it is found that the large mount of noise in the detected sonogram data always impose a negative effect upon reconstruction, and such a negative effect might run through the whole iterative process. Bayesian reconstruction can greatly improve reconstruction by incorporating image prior information. But we also find that, heavily relied on the information within a limited neighborhood, conventional Bayesian methods can only contribuite limit local prior information to reconstruction. On one hand, the smoothing QM (Quadratic Membrane) smoothing prior tends to produce an unfavorable oversmoothing effect, and on the other hand the edge-preserving nonquadratic prior might also bring staircase edge artifact to reconstruction.
Our work on PET reconstruction algorithm is based on how to further accelerate the convergence rate of reconstruction algorithm and how to further improve the quality for reconstructions.We have done following work on PET reconstruction algorithms:
1, proposeing a new fast OSCG (Ordered Subsets Conjugate Gradient) algorithm, which combines the idea of OS (Odered Subset) and CG method to further accelerate the rate for PET reconstruction.
2, proposing a novel coupled feedback (CF) iterative model and the relevant reconstruction algorithm. The new methods can relieve the negative effect of the noise in detected sinogram data on PET reconstruction by an iterative sinogram-correcting method.
3, proposing a quadratic hybrid multi-order (QHM) prior model that combine both QM prior and QP prior effectively, the QHM prior is capable of facilitating an adaptive utilization of QM prior and QP prior in PET reconstruction.
4, proposing a novel nonlocal Markov Random Fields (MRF) prior with quadratic prior energy form, which is able to exploit global image prior information and provides more effective regularization for PET reconstruction.
In experiments, we apply above four approaches in emission tomography and transmission tomography in PET. Relevant experimentations and analyses show that the four new algorithms, which are all based on MRF and optimation theory, are all able to improve currently PET reconstruction to different extents, respectively
最大似然—期望最大法(Maximum-Likelihood Expectation-Maximization，ML-EM)能够针对系统模型的物理效应和探测数据和噪声的统计泊松特性建立数学模型，其重建的图像质量要优于传统的FBP方法。然而，单纯的传统ML-EM方法依然有两个缺点：1，在迭代过程中，会产生质量退化的图像而导致的棋盘效应，从而导致非收敛的迭代过程；2，收敛速度较慢，需要大量迭代次数才能重建出可以接受的图像。
近二十年来，国际上出现了很多解决以上两个问题的方法。一方面，基于马尔可夫随机场(Markov Random Fields，MRF)的贝叶斯(Bayesian)重建方法或者最大化后验估计(Maximum A Posteriori，MAP)的方法通过引入正则化项来引进目标同位素密度数据在空间上的概率分布的先验信息，能够明显改善重建图像质量以及迭代过程的收敛性，该方法已被证明了其在理论上的正确性和实际上的有效性。另一方面，很多科学家也提出一些有效的方法和算法来解决ML-EM方法收敛速度慢的问题，1994年，H．Malcolm Houdson和Richard S．Larkin提出使用有序子集(Ordered Subsets，OS)的方法将完整探测数据sinogram分割成有序部分数据的方法来减小每次迭代的运算量，从而达到了加快收敛速度的目的；密歇根大学的J．A．Fessler小组提出了能够提高收敛速度的SAGE(Space-Alternating Generalized EM)收敛重建算法和PSCD(Paraboloidal Surrogate Coordinate Ascent)收敛重建算法；另外，Erkan (?) Mumcuoolu，David S．Lalush及Fessler也分别提出将共轭梯度法(Conjugate Gradient，CG)应用于PET重建算法中以加快收敛速度。
对于重建速度的问题，虽然能够反映探测数据统计特性的迭代重建能够重建出较传统的FBP重建更好的图像，然而其应用一直受到重建速度慢的缺点的限制；在重建图像质量方面，作者发现，sinogram探测值中大量的噪声一直对重建有着很大的负面影响，而且此种负面影响持续贯穿于整个在迭代过程中。虽然引入了图像先验信息的Bayesian能够在很大程度上改善迭代重建，但是依赖于传统的局部邻域信息的Bayesian方法只能为重建提供有限的局部先验信息，一方面，传统的平滑二次QM(Quadratic Membrane)先验易在重建的结果图像中产生过平滑效果，另一方面，具有边缘保持作用的非二次先验则会给重建图像带来阶梯状的边缘伪影。
本文对于PET重建算法的研究工作同样也是基于如何进一步加快重建算法的收敛速度和进一步提高重建图像的质量。作者做了以下四项PET重建算法方面的工作：
1，提出将有序子集的思想和共轭梯度法相结合的新的快速子集共轭梯度重建算法(OSCG Ordered Subsets Conjugate Gradient)来加快PET重建的收敛速度。
2，提出基于修改sinogram探测值的新的耦合反馈(Coupled Feedback，CF)迭代模型来降低在重建迭代中探测数据sinogram中的噪声对PET重建的负面影响。
3，提出一种新的有效的综合了QM先验和QP(Quadratic Plate)先验的MRF二次Membrane-Plate混合先验模型，实现了在Bayesian重建中根据目标图像中各点的性质来自适应的选择QM先验和QP先验。
4，提出一个新的且具有二次先验能量形式的MRF非局部(nonlocal)先验模型，实现了利用图像中的全局的信息来为PET重建提供更加有效的正则化作用。
试验中将以上四种方法应用于相应的PET断层发射成像(Emission Tomography)和断层透射成像(Transmission Tomography)，相关试验分析表明：本文所提出的四种基于MRF和优化理论的四种新算法均能够分别在不同程度上改善PET重建。
As an effective functional imaging method for medical images, positron emission tomography (PET) is able to represent heart and brain metabolism and functions on molecule level by imaging techniques, and has shown great performance in oncology, cardiopathy, neurology and new medicine studies. But positron emission tomography is an ill-posed inverse problem because the observed projection data are contaminated by noise due to low count rate and physical effects. Though needing less compution cost, traditional filter back projection (FBP) method often reconstruct noisy images of low quality.
Better expressing system models of physical effects and modeling the statistical poisson character of the data, the famous maximum-likelihood expectation-maximization (ML-EM) approach outperforms the FBP method with regard to image quality. However, pure traditional ML-EM approach is still notorious for two drawbacks: 1, the reconstructed images always start deteriorating to produce "checkerboard effect" as the iteration proceeds; 2, ML-EM approach suffers slow convergence, and a large number of iterations are needed to be performed before obtaining acceptable reconstructions.
Many methods have been proposed to overcome above two frawbacks in the past twenty years. On one hand, Bayesian methods or equivalently MAP (Maximum A Posteriori) methods, which incorporate MRF prior information of objective isotope density data into the ML-EM algorithm through regularization or prior terms, have been proved theoretically correct and practically effective compared to other methods. Compared to traditional ML-EM algorithm, Bayesian reconstruction shows a better performance in both improving convergence behavior and producing more appealing images. On the other hand, as to the drawback of slow convergence for ML-EM algorithm, some effective solutions and algorithms have also been proposed. In 1994, H. Malcolm Houdson and Richard S. Larkin proposed segementing the original whole sonogram data into several ordered data by OS (Ordered Subsets) method to lower the compution cost of each iteration step; J. A. Fessler and his group proposed fast convergent SAGE (Space-Alternating Generalized EM) algorithm and PSCD (Paraboloidal Surrogate Coordinate Ascent) algorithm for PET. And Erkan (?) Mumcuoglu, David S. Lalush and Fessler also applied CG (Conjugate Gradient) in PET reconsruction algorithm to obtain fast convergence rate.
As to the problem of reconstruction rate, the iterative methods which incorporate the statistical characters of scaned data, although able to reconstruct better images than conventional FBP approaches, their applications are also hindered by the slow reconstruction rate. As to the problem of reconstructed image quality, it is found that the large mount of noise in the detected sonogram data always impose a negative effect upon reconstruction, and such a negative effect might run through the whole iterative process. Bayesian reconstruction can greatly improve reconstruction by incorporating image prior information. But we also find that, heavily relied on the information within a limited neighborhood, conventional Bayesian methods can only contribuite limit local prior information to reconstruction. On one hand, the smoothing QM (Quadratic Membrane) smoothing prior tends to produce an unfavorable oversmoothing effect, and on the other hand the edge-preserving nonquadratic prior might also bring staircase edge artifact to reconstruction.
Our work on PET reconstruction algorithm is based on how to further accelerate the convergence rate of reconstruction algorithm and how to further improve the quality for reconstructions.We have done following work on PET reconstruction algorithms:
1, proposeing a new fast OSCG (Ordered Subsets Conjugate Gradient) algorithm, which combines the idea of OS (Odered Subset) and CG method to further accelerate the rate for PET reconstruction.
2, proposing a novel coupled feedback (CF) iterative model and the relevant reconstruction algorithm. The new methods can relieve the negative effect of the noise in detected sinogram data on PET reconstruction by an iterative sinogram-correcting method.
3, proposing a quadratic hybrid multi-order (QHM) prior model that combine both QM prior and QP prior effectively, the QHM prior is capable of facilitating an adaptive utilization of QM prior and QP prior in PET reconstruction.
4, proposing a novel nonlocal Markov Random Fields (MRF) prior with quadratic prior energy form, which is able to exploit global image prior information and provides more effective regularization for PET reconstruction.
In experiments, we apply above four approaches in emission tomography and transmission tomography in PET. Relevant experimentations and analyses show that the four new algorithms, which are all based on MRF and optimation theory, are all able to improve currently PET reconstruction to different extents, respectively
引文
[1] 潘中允主编．PET诊断学[M]．北京：人民卫生出版社．2005．
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